for all \(x\)  and \(y\)  in \({X}\) ; it satisfies the Upper Weak Parallelogram property (UWP) if for some (possibly different) constant and exponent the reverse inequality holds for all \(x\)  and \(y\)  in \({X}\) . If \({X}\)  is a Hilbert space, then the parallelogram law holds, corresponding to \(r=2\)  and \(C=1\) . Otherwise, these inequalities generalize Clarkson's inequalities [1]}, and the parameters \(r\)  and \(C\)  give a sense of how far the space \({X}\)  departs from behaving like a Hilbert space.
